Nonexistence of invariant manifolds in fractional-order dynamical systems

Patil, Madhuri

doi:10.1007/s11071-020-06073-9

Cited by 3 publications

(1 citation statement)

References 49 publications

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“…The exception to this rule appears when the system contains fractional derivatives [18]. There is also the opposite procedure, in which the use of Caputo derivatives stabilizes the chaotic behavior [19], for example, in optimal control problems. One of the basic reasons for the introduction of new fractional derivatives consists in building operators that preserve the history of interactions.…”

Section: Discussionmentioning

confidence: 99%

Fractional Complex Euler–Lagrange Equation: Nonconservative Systems

Toma,

Postavaru

2023

Fractal Fract

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Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system.

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Section: Discussionmentioning

confidence: 99%

Fractional Complex Euler–Lagrange Equation: Nonconservative Systems

Toma,

Postavaru

2023

Fractal Fract

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Controlling fractional difference equations using feedback

Joshi

Gade

2023

Chaos, Solitons & Fractals

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Study of low-dimensional nonlinear fractional difference equations of complex order

Joshi

Gade

2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study the fractional maps of complex order, [Formula: see text], for [Formula: see text] and [Formula: see text] in one and two dimensions. In two dimensions, we study Hénon, Duffing, and Lozi maps, and in [Formula: see text], we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in [Formula: see text] can be done in two different ways, which are not equivalent for fractional order and lead to different bifurcation diagrams. We observed that the smooth maps, such as logistic, Gauss, Duffing, and Hénon maps, do not show chaos, while discontinuous maps, such as Bernoulli and circle maps,show chaos. The tent and Lozi map are continuous but not differentiable, and they show chaos as well. In [Formula: see text], we find that the complex fractional-order maps that show chaos also show multistability. Thus, it can be inferred that the smooth maps of complex fractional order tend to show more regular behavior than the discontinuous or non-differentiable maps.

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Nonexistence of invariant manifolds in fractional-order dynamical systems

Cited by 3 publications

References 49 publications

Fractional Complex Euler–Lagrange Equation: Nonconservative Systems

Fractional Complex Euler–Lagrange Equation: Nonconservative Systems

Controlling fractional difference equations using feedback

Study of low-dimensional nonlinear fractional difference equations of complex order

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